On the properties of Generalized Jacobsthal and Generalized
Jacobsthal-Lucas sequences
SUKANYA SOMPROM, PIYANUT PUANGJUMPA, ARUNTIDA SICHIANGHA
Department of Mathematic, Faculty of Science and Technology,
Surindra Rajabhat University, Surin 32000
THAILAND
Abstract: In this paper, we study the Generalized Jacobsthal and Generalized Jacobsthal-Lucas sequences. We
also present some properties on relationship between the Generalized Jacobsthal sequence and Generalized
Jacobsthal-Lucas sequence by using Binet’s formula for derivation. In addition, we showed colollaries and exm-
ples of Jacobsthal and Jacobsthal-Lucas sequence.
Key-Words: Generalized Jacobsthal sequence, Generalized Jacobsthal-Lucas sequence , Binet’s formula.
Received: April 19, 2023. Revised: July 23, 2023. Accepted: August 16, 2023. Published: September 20, 2023.
1 Introduction
In the last years, many researchers have been in-
terested in studying the sequence of numbers and
recurrence relation, for example, Fibonacci, Lucas,
Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas, etc. It
can also be applied in applied mathematics, number
theory, computers, etc. Also, many applications in di-
verse fields such as finance, architectural science, art,
petals on flower, arrangement of seeds on flower etc.,
[1]. The Jacobsthal and Jacobsthal-Lucas sequences,
[2], is defined by the recurrence relation, respectively,
Jn=Jn−1+2Jn−2and jn=jn−1+2jn−2(1)
for n≥2where J0= 0, J1= 1 and j0= 2, j1= 1.
The Binet’s formulas, respectively, for Jacobsthal
and Jacobsthal-Lucas sequences is given by
Jn=rn
2−rn
1
r2−r1
and jn=r2
1+r2
2
where r1= 2 and r2=−1are the root of the charac-
teristic equation
x2−x−2 = 0.
For equation (1), the first Jacobsthal and Jacobsthal–
Lucas sequences are written as, respectively,
Jn={0,1,1,3,5,11,21,43,85,171, ...}
and
jn={2,1,5,7,17,31,65,127,257,511, ...}.
In 2014, [3], studied the product and sum of the
Jacobsthal and Jacobsthal–Lucas sequences by using
the explicit Binet’s formulas to show their prop-
erties. In 2008, [4], presented some properties of
the Jacobsthal-Lucas E-matrix and R-matrix by the
Matrix Method. In 2014, [5], derive various formula
for sum of k-Jacobsthal numbers with indexes in an
arithmetic sequence. In 2016, [6], presented some
properties of k-Jacobsthal and k-Jacobsthal–Lucas
sequences and some relationship of these sequence
by using the Binet’s formulas. Moreover, many
researchers are also interested in some properties
of Jacobsthal-like, (s, t)-Jacobthal, (s, t)-Jacobthal-
Lucas etc. by using Binet’s formula or Matrix
Method for derivation see more details in [7], [8].
Based on extensive research on the properties of
Jacobsthal and Jacobsthal-Lucas numbers, it was dis-
covered that additional properties could be derived,
and the results of the study could be extended in
the form of generalized Jacobsthal and generalized
Jacobsthal-Lucas sequences. The proof will use the
Binet’s formula.
Next, in sections 2, 3 and 4, we discuss some basic
facts, definitions, proven theorems, corollaries, pro-
viding examples and conclusions.
2 Preliminaries
In section 2, we recall some definitions of the
Generalized Jacobsthal sequence and Generalized
Jacobsthal-Lucas sequences.
Definition 2.1, [10]. Let J(k, n)be the Gener-
alized Jacobsthal sequence defined recursively as
follows :
J(k, n) = (k−1) J(k, n −1) + kJ (k, n −2) ,
for n≥2with J(k, 0) = 0 and J(k, 1) = 1. That
is, from relationship of the Generalized Jacobsthal
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Sukanya Somprom, Piyanut Puangjumpa, Aruntida Sichiangha
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sequence, we can write the first Generalized Jacob-
sthal sequence for case 0≤n≤7as follows :
J(k, 0) = 0, J (k, 1) = 1, J (k, 2) = k−1,
J(k, 3) = k2−k+ 1, J (k, 4) = k3−k2+k−1,
J(k, 5) = k4−k3+k2−k+ 1,
J(k, 6) = k5−k4+k3−k2+k−1,
J(k, 7) = k6−k5+k4−k3+k2−k+ 1.
Next, Table 1 shows the first few terms of the se-
quences J(k, n)for case 2≤k≤10 and 2≤n≤8,
Table 1: Generalized Jacobsthal sequence.
J(k, n)2 3 4 5 6 7 8
2 1 3 5 11 21 43 85
3 2 7 20 61 182 547 1640
4 3 13 51 205 819 3277 13107
5 4 21 104 521 2604 13021 65104
6 5 31 185 1111 6665 39991 239945
7 6 43 300 2101 14706 102943 720600
8 7 57 455 3641 29127 233017 1864135
9 8 73 656 5905 53144 478297 4304672
10 9 91 909 9091 90909 909091 9090909
Definition 2.2, [10]. Let j(k, n)be the Generalized
Jacobsthal-Lucas sequence defined recursively as fol-
lows :
j(k, n) = (k−1) j(k, n −1) + kj (k, n −2) ,
for n≥2with j(k, 0) = 2 and j(k, 1) = 1. That
is, from relationship of the Generalized Jacobsthal-
Lucas sequence, we can write the first Generalized
Jacobsthal-Lucas sequence for case 0≤n≤7as
follows :
j(k, 0) = 2, j (k, 1) = 1, j (k, 2) = 3k−1,
j(k, 3) = 3k2−3k+1, j (k, 4) = 3k3−3k2+3k−1,
j(k, 5) = 3k4−3k3+ 3k2−3k+ 1,
j(k, 6) = 3k5−3k4+ 3k3−3k2+ 3k−1,
j(k, 7) = 3k6−3k5+ 3k4−3k3+ 3k2−3k+ 1.
Next, Table 2 shows the first few terms of the
sequences j(k, n)for case 2≤k≤10 and
2≤n≤8,
Table 2: Generalized Jacobsthal-Lucas sequence.
j(k, n)2 3 4 5 6 7 8
2 5 7 17 31 65 127 257
3 8 19 62 181 548 1639 4922
4 11 37 155 613 2459 9829 39323
5 14 61 314 1561 7814 39061 195314
6 17 91 557 3331 19997 119971 719837
7 20 127 902 6301 44120 308827 2161802
8 23 169 1367 10921 87383 699049 5592407
9 26 217 1970 17713 159434 1434889 12914018
10 29 271 2729 27271 272729 2727271 27272719
Let r1and r2be the roots of the characteristic equation
r2−(k−1)r−k= 0.
Since △= (k+ 1)2>0for k≥2, we have
r1=−1and r2=k.
They satisfy the following equations,
r1+r2=k−1, r2−r1=k+ 1 and r1r2=−k.
For n≥0, k ≥2. The Binet’s formula, respec-
tively, for Generalized Jacobsthal and Generalized
Jacobsthal-Lucas sequence is given by
J(k, n) = rn
2−rn
1
r2−r1
and j(k, n) = c1r2
1+c2r2
2
where c1=2k−1
k+ 1 and c2=3
k+ 1 . Particular
cases of the previous definition are :
•If k= 2 and J0= 0 , J1= 1 , the Jacobsthal
sequence is obtained.
•If k= 2 and j0= 2 , j1= 1, the Jacobsthal-Lucas
sequence is obtained.
Researchers have been talking about the prop-
erties of Generalized Jacobsthal and Generalized
Jacobsthal-Lucas sequences. For example, in 2021,
[9], introduced the Generalization of Jacobsthal and
Jacobsthal-Lucas numbers and gave the properties of
related matrix and the sum of terms of the sequences.
In 2022, [10], studied the Generalization of Jacob-
sthal and Jacobsthal-Lucas numbers and give gen-
erating functions, Binet’s formulas for these num-
bers. They show properties generalize the well-
known results for classical Jacobstal numbers and
Jacobsthal-Lucas numbers. Moreover, many authors
have studied and presented some important relation-
ships between Generalized Jacobsthal and General-
ized Jacobsthal–Lucas sequences. For example, in
2005, [11], definined and proved theorems and corol-
laries of the incomplete generalized Jacobsthal and
incomplete generalized Jacobsthal-Lucas number. In
addition, in 2015, [12], presented new families and
generating functions for generalized Jacobsthal and
the Jacobsthal-Lucas sequences.
The purpose of this paper is to establish the rela-
tionship between generalized Jacobsthal and general-
ized Jacobsthal-Lucas sequences in terms of the prod-
uct and the sum of both numbers, expressing them
in real number form and the generalized Jacobsthal
number format using the Binet’s formula. From the
results, it is found that the properties we have proven
also hold true for Jacobsthal and Jacobsthal-Lucas se-
quences when considering the case where k= 2.
3 Main Theorem
In this section, we gave some theorem of the Gener-
alized Jacobsthal and Generalized Jacobsthal-Lucas
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sequences. Moreover, we presented corollaries and
examples as follows.
Theorem 3.1. Let k, n, s be integers. For k≥2and
n≥s≥1. Then
J(k, 2n)J(k, 2s)−J2(k, n +s)
=−k2sJ2(k, n −s).(2)
Proof. By using Binet’s formula, we have
J(k, 2n)J(k, 2s)−J2(k, n +s)
=r2n
2−r2n
1r2s
2−r2s
1
(r2−r1)2−rn+s
2−rn+s
12
(r2−r1)2
=r2n+2s
2−r2s
1r2n
2−r2n
1r2s
2+r2n+2s
1
(r2−r1)2
−r2n+2s
2−2 (r1r2)n+s+r2n+2s
1
(r2−r1)2
=2 (r1r2)n+s−r2s
1r2n
2−r2n
1r2s
2
(r2−r1)2
=(r1r2)n+s
(r2−r1)22−r1
r2n−s
−r2
r1n−s
=(r1r2)n+s
−(r1r2)n−s(r2−r1)2
×r2(n−s)
1−2 (r1r2)n−s+r2(n−s)
2
=−(r1r2)2srn−s
2−rn−s
1
r2−r12
= (−1) k2sJ2(k, n −s)
=−k2sJ2(k, n −s).
Thus, the proof is complete.
From Theorem 3.1, we considered the equation
(2) by choosing n= 3 and s= 1 as an example.
It was found that on the left side of equation (2), it
could be calculated as −k4+ 2k3−k2. And on the
right, it could also be calculated as −k4+ 2k3−k2,
where both sides were equal. Based on the calculated
values from Table 1, when we considered the case
where k= 3, we found that both sides resulted in the
same value, which is −36.
Theorem 3.2. Let k, n, s be integers and
k≥2, n ≥s≥1. Then, we have
j(k, 2n)j(k, 2s)−j2(k, n +s)
= (6k−3)k2sJ2(k, n −s).(3)
Proof. By using Binet’s formula, we have
j(k, 2n)j(k, 2s)−j2(k, n +s)
=c1r2n
1+c2r2n
2c1r2s
1+c2r2s
2
−c1rn+s
1+c2rn+s
22
=c2
1r2n+2s
1+c1c2r2n
1r2s
2+c1c2r2s
1r2n
2+c2
2r2n+2s
2
−c2
1r2n+2s
1−2c1c2rn+s
1rn+s
2−c2
2r2n+2s
2
=c1c2r2n
1r2s
2+c1c2r2s
1r2n
2−2c1c2rn+s
1rn+s
2
=c1c2(r1r2)n+sr1
r2n−s
−2 + r2
r1n−s
=(c1c2) (r1r2)n+s
(r1r2)n−sr2(n−s)
1−2 (r1r2)n−s+r2(n−s)
2
= (c1c2) (r1r2)2srn−s
2−rn−s
12
= (c1c2) (r1r2)2s(r2−r1)2rn−s
2−rn−s
1
r2−r12
=2k−1
1 + k 3
1 + k(−k)2s(k+ 1)2J2(k, n −s)
= (6k−3)k2sJ2(k, n −s).
Thus, the proof is complete.
From Theorem 3.2, we considered the equation
(3) by choosing n= 3 and s= 1 as an example.
It was found that on the left side of equation (3), it
could be calculated as 6k5−15k4+ 12k3−3k.
And on the right, it could also be calculated as
6k5−15k4+ 12k3−3k, where both sides were
equal. Based on the calculated values from Table
1 and Table 2, when we considered the case where
k= 3, we found that both sides resulted in the same
value, which is 540.
Next, from the above theorem, we obtained the
well-known identities for Generalized Jacobsthal and
Generalized Jacobsthal-Lucas sequence. For k= 2,
i.e. J(2, n) = Jnand j(2, n) = jn, Theorem 3.1
and Theorem 3.2 reduces to new identities in the
following corollary.
Corollary 3.3. Let n, s be integer. For n≥s,
we have
J2nJ2s−J2
n+s=−22sJ2
n−s(4)
Moreover,
j2nj2s−j2
n+s= 9 (2)2sj2
n−s.(5)
From Corollary 3.3, we considered the Table 1
and Table 2, by choosing n= 3 and s= 2 as an
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example. It was found that on the left side of equation
(4), it could be calculated as -16. On the right side,
it could be calculated as -16. It was found that both
were equal. Similarly, for n= 3 and s= 2 it was
found that on the left and right side of equation (5),
it could be calculated as 144. It was found that both
were equal.
Theorem 3.4. Let k, n be integers and k≥2, n ≥1.
Then, we have
kJ2(k, n −1) + (k−1) J(k, n −1) J(k, n)
−J2(k, n) = (−1)nkn−1.(6)
Proof. Since,
J(k, n) = (k−1) J(k, n −1) + kJ (k, n −2),
we have
J2(k, n) = (k−1) J(k, n −1) J(k, n)
+kJ (k, n −2) J(k, n)
i.e.,
(k−1) J(k, n −1) J(k, n)−J2(k, n)
=−kJ (k, n −2) J(k, n).
By using Binet’s formula, we have
kJ2(k, n −1) + (k−1) J(k, n −1) J(k, n)
−J2(k, n)
=kJ2(k, n −1) −kJ (k, n −2) J(k, n)
=krn−1
2−rn−1
12
(r2−r1)2−rn−2
2−rn−2
1(rn
2−rn
1)
(r2−r1)2
=k
(r2−r1)2r2(n−1)
2−2rn−1
1rn−1
2+r2(n−1)
1
−k
(r2−r1)2r2(n−1)
2−rn
1rn−2
2−rn−2
1rn
2+r2(n−1)
1
=k
(r2−r1)2−2 (r1r2)n−1+rn
1rn−2
2+rn−2
1rn
2
=k(r1r2)n−1
(r2−r1)2−2 + r1
r2
+r2
r1
=k(−k)n−1
(r2−r1)2−2r1r2+r2
1+r2
2
r1r2
=k(−k)n−1
(r2−r1)2(r2−r1)2
−k
=−(−k)n−1
= (−1)nkn−1.
Thus, the proof is complete.
From Theorem 3.4, we considered the equation
(6) by choosing n= 3 as an example. It was found
that on the left side of equation (6), it could be
calculated as −k2. And on the right, it could also
be calculated as −k2, where both sides were equal.
Based on the calculated values from Table 1, when
we considered the case where k= 3, we found that
both sides resulted in the same value, which is −9.
Theorem 3.5. Let k, n be integers and k≥2, n ≥1.
Then, we have
kj2(k, n −1) + (k−1) j(k, n −1) j(k, n)
−j2(k, n) = (−k)n−1(6k−3) .(7)
Proof. Since,
j(k, n) = (k−1) j(k, n −1) + kj (k, n −2),
we have
j2(k, n) = (k−1) j(k, n −1) j(k, n)
+kj (k, n −2) j(k, n)
i.e.,
(k−1) j(k, n −1) j(k, n)−j2(k, n)
=−kj (k, n −2) j(k, n).
By using Binet’s formula, we have
j2(k, n −1) + (k−1) j(k, n −1) J(k, n)
−j2(k, n)
=j2(k, n −1) −kj (k, n −2) j(k, n)
=kc1rn−1
1+c2rn−1
22
−kc1rn−2
1+c2rn−2
2(c1rn
1+c2rn
2)
=kc2
1r2(n−1)
1+ 2c1c2(r1r2)n−1+c2
2r2(n−1)
2
−kc2
1r2(n−1)
1+c1c2rn−2
1rn
2
−kc1c2rn
1rn−2
2+c2
2r2(n−1)
2
=k2c1c2(r1r2)n−1−c1c2rn−2
1rn
2−c1c2rn
1rn−2
2
=k(c1c2) (r1r2)n−12−r2
r1−r1
r2
=k(c1c2) (r1r2)n−12r1r2−r2
2−r2
1
r1r2
=k(c1c2) (r1r2)n−22r1r2−r2
2−r2
1
=−k(c1c2) (r1r2)n−2(r2−r1)2
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=−k(c1c2) (−k)n−2(k+ 1)2
= (−k)n−12k−1
k+ 1 ·3
k+ 1(k+ 1)2
= (−k)n−1(6k−3) .
Thus, the proof is complete.
From Theorem 3.5, we considered the equation (7)
by choosing n= 3 as an example. It was found that
on the left side of equation (7), it could be calculated
as 6k3−3k2. And on the right, it could also be cal-
culated as 6k3−3k2, where both sides were equal.
Based on the calculated values from Table 2, when
we considered the case where k= 3, we found that
both sides resulted in the same value, which is 135.
Next, from the above theorem, we obtained the
well-known identities for Generalized Jacobsthal
and Generalized Jacobsthal-Lucas sequences. For
k= 2, i.e. J(2, n) = Jnand j(2, n) = jn, Theorem
3.4 and Theorem 3.5 reduce to new identities in the
following corollary.
Corollary 3.6. For nbe integer, we have
2J2
n−1+Jn−1Jn−J2
n= (−1)n2n−1(8)
Moreover,
2j2
n−1+jn−1jn−j2
n= 9 (−2)n−1. (9)
From Corollary 3.6, we considered the Table 1
and Table 2, by choosing n= 3 as an example.
It was found that on the left side of equation (8),
it could be calculated as −4. On the right side, it
could be calculated as −4. It was found that both
were equal. Similarly, for n= 3, it was found that
on the left and right side of equation (9), it could
be calculated as 36. It was found that both were equal.
Theorem 3.7. Let k, n, s be integers and
k≥2, n ≥s≥1. Then, we have
J(k, n)j(k, s + 1) −j(k, n + 1) J(k, s)
= (−k)sJ(k, n −s).(10)
Proof. By using Binet’s formula, we have
J(k, n)j(k, s + 1) −j(k, n + 1) J(k, s)
=rn
2−rn
1
r2−r1c1rs+1
1+c2rs+1
2
−c1rn+1
1+c2rn+1
2rs
2−rs
1
r2−r1
=c1rs+1
1rn
2+c2rn+s+1
2−c1rn+s+1
1−c2rn
1rs+1
2
r2−r1
−c1rn+1
1rs
2−c1rn+s+1
1+c2rn+s+1
2−c2rs
1rn+1
2
r2−r1
=(rs
1rn
2) (c1r1+c2r2)
r2−r1
−(rn
1rs
2) (c1r1+c2r2)
r2−r1
=c1r1+c2r2
r2−r1(rs
1rn
2−rn
1rs
2)
= (c1r1+c2r2) (r1r2)srn−s
2−rn−s
1
r2−r1
=j(k, 1) (−k)sJ(k, n −s)
= (−k)sJ(k, n −s).
Thus, the proof is complete.
From Theorem 3.7, we considered the equation
(10) by choosing n= 3 and s= 2 as an example.
It was found that on the left side of equation (10), it
could be calculated as −k2. And on the right, it could
also be calculated as −k2, where both sides were
equal. Based on the calculated values from Table
1 and Table 2, when we considered the case where
k= 3, we found that both sides resulted in the same
value, which is −9.
Theorem 3.8. Let k, n, s be integers and
k≥2, n ≥s≥1. Then, we have
j(k, n)J(k, s + 1) −J(k, n + 1) j(k, s)
= (−1)s+1 (2k−3) ksJ(k, n −s).(11)
Proof. By using Binet’s formula, we have
j(k, n)J(k, s + 1) −J(k, n + 1) j(k, s)
= (c1rn
1+c2rn
2)rs+1
2−rs+1
1
r2−r1
−rn+1
2−rn+1
1
r2−r1(c1rs
1+c2rs
2)
=c1rn
1rs+1
2+c2rn+s+1
2−c1rn+s+1
1−c2rs+1
1rn
2
r2−r1
−c1rs
1rn+1
2−c1rn+s+1
1+c2rn+s+1
2−c2rn+1
1rs
2
r2−r1
=c1r2(rn
1rs
2)−c2r1(rs
1rn
2)
r2−r1
−c1r2(rs
1rn
2)−c2r1(rn
1rs
2)
r2−r1
=1
r2−r1
(c1r2+c2r1) (rn
1rs
2−rs
1rn
2)
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=1
r2−r1
(c1r2+c2r1) (r1r2)srn
1
rs
1
−rn
2
rs
2
=−1
r2−r1
(c1r2+c2r1) (r1r2)srn−s
2−rn−s
1
=−(c1r2+c2r1) (−k)sJ(k, n −s)
= (−1)s+1 (2k−3) ksJ(k, n −s).
Thus, the proof is complete.
From Theorem 3.8, we considered the equation
(11) by choosing n= 3 and s= 2 as an example.
It was found that on the left side of equation (11), it
could be calculated as −2k3+ 3k2. And on the right,
it could also be calculated as −2k3+ 3k2, where both
sides were equal. Based on the calculated values
from Table 1 and Table 2, when we considered the
case where k= 3, we found that both sides resulted
in the same value, which is −27.
Corollary 3.9. Let n≥sbe integers. Then,
we have
Jnjs+1 −jn+1Js= (−2)sJn−s(12)
Moreover,
jnJs+1 −Jn+1js= (−1)s+1 (2)sJn−s.(13)
From Corollary 3.9, we considered the Table 1 and
Table 2, by choosing n= 3 and s= 2 as an example.
It was found that on the left side of equation (12), it
could be calculated as 4 . On the right side, it could
be calculated as 4. It was found that they both were
equal. Similarly, by choosing n= 3 and s= 2 as an
example. It was found that on the left side of equation
(13), it could be calculated as -4. On the right side, it
could be calculated as -4. It was found that both were
equal.
4 Conclusion
We have studied the definitions of the General-
ized Jacobsthal and Generalized Jacobsthal-Lucas se-
quences. From the study, we have discovered some
important properties of those relationships. In the
proof, we will use the Binet form. Moreover, for the
properties that we have discovered, if we consider
the case where k= 2, we will obtain the proper-
ties that cover the relationships of the Jacobsthal and
Jacobsthal-Lucas sequences. Furthermore, we have
also presented corollaries and demonstrated some ex-
amples to support the properties that we have studied.
In the future, we are interested in studying the rela-
tionships of the Generalized Jacobsthal and General-
ized Jacobsthal-Lucas sequences in the form of ma-
trices or defining new relationships of these numbers.
In addition, we aim to find some specific properties
of these numbers as well.
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.70
Sukanya Somprom, Piyanut Puangjumpa, Aruntida Sichiangha
E-ISSN: 2224-2880
639
Volume 22, 2023
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Sukanya Somprom is responsible for the con-
ceptualization of the research problem, formal
analysis and the supervision of the work.
Piyanut Puangjump is responsible for
formal analysis and validation.
Aruntida Sichiangha is responsible for the
formal analysis and corresponding author.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflicts of Interest
The authors have no conflicts of interest to
declare that are relevant to the content of this
article.
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This article is published under the terms of the
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.70
Sukanya Somprom, Piyanut Puangjumpa, Aruntida Sichiangha
E-ISSN: 2224-2880
640
Volume 22, 2023
